Problem: Simplify and expand the following expression: $ \dfrac{a}{5a + 8}+\dfrac{2a}{5a + 4} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5a + 8)(5a + 4)$ Multiply the first term by $\dfrac{5a + 4}{5a + 4}$ $ \begin{align*} \dfrac{a}{5a + 8} \times \dfrac{5a + 4}{5a + 4} & = \dfrac{(a)(5a + 4)}{(5a + 8)(5a + 4)} \\ & = \dfrac{5a^2 + 4a}{(5a + 8)(5a + 4)}\end{align*} $ Multiply the second term by $\dfrac{5a + 8}{5a + 8}$ $ \begin{align*} \dfrac{2a}{5a + 4} \times \dfrac{5a + 8}{5a + 8} & = \dfrac{(2a)(5a + 8)}{(5a + 4)(5a + 8)} \\ & = \dfrac{10a^2 + 16a}{(5a + 4)(5a + 8)}\end{align*} $ Now we have: $ = \dfrac{5a^2 + 4a}{(5a + 8)(5a + 4)} + \dfrac{10a^2 + 16a}{(5a + 4)(5a + 8)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5a^2 + 4a + 10a^2 + 16a}{(5a + 8)(5a + 4)} $ $ = \dfrac{15a^2 + 20a}{(5a + 8)(5a + 4)}$ Expand the denominator: $ = \dfrac{15a^2 + 20a}{25a^2 + 60a + 32}$